# Visualizing Quantum Mechanics with Mathematica

*By Timothy Atherton On December 21, 2012 · Leave a Comment*

Last semester, I saw a colloquium by Judah Schwartz from the Education department at Tufts University. Judah was one of the co-authors of a classic paper Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena from 1967. He and his co-authors used an IBM 7094 and a CDC 3600 to numerically solve the Time Dependent Schrödinger Equation for one electron in one dimension. They made movies of how wave packets behave in some common textbook physics problems, such as the potential barrier for which a few frames of which are illustrated on the left. The production of these movies was painstaking: the computer plotted the result from a single time step on a cathode ray tube, and the display was photographed to produce each frame. Hence, these movies represent a kind of stop motion animation. They’re historically highly significant as they represent one of the first times computerized visualization was used to try to understand quantum mechanics—and even more so because they’re one of the first uses of computer graphics for educational purposes. In many ways they foreshadow the amazing potential of programs like *Mathematica* to enhance education.

There’s an incredible amount of detail embedded in the videos such as the transient “ringing” as the wave packet actually hits the barrier. Traditional undergraduate physics education tends to de-emphasize the Time Dependent Schrödinger Equation because it’s so difficult to solve; instead we spend a lot of time looking at the Time-Independent Schrödinger Equation instead which is mathematically much more tractable. Another appealing feature about the movies is that the nature of the constraints of the early computers used to make them resulted in a visual design that’s strikingly minimalistic, information rich and very elegant—I think Edward Tufte would approve!

Even so, they suffer from an obvious problem: wouldn’t it be great if they were interactive? We know from education research like the PhET project that structuring open-ended activities around interactive simulations can be a great way for students to learn about quantum mechanics. Rather than seeing a movie, wouldn’t it be great if they could change the potentials and wave packets and see what happens?

Thanks to Moore’s law, a problem that in 1967 required a supercomputer is now doable on your cellphone. Now we have tools like *Mathematica, *which I already use extensively in my physics classes. Could I make a modern, interactive equivalent of Judah’s amazing movies that I could use in my Physics 13 Modern Physics class—which is the first time students ever see Quantum Mechanics? And moreover, could I preserve the beautiful design style of the originals?

Here’s my effort: download the Mathematica 9 notebook containing the program. I used it to great success in my class, both in lectures and in homeworks—Quantum Mechanics is extremely difficult to learn because it requires one to reason about things for which one lacks an intuition from everyday experience. The movies help because they allow students to develop such an intuition. They make the transition from the classical to the quantum world just a little easier. I hope you find the program useful!

Here are some sample movies generated with the program:

*Wavepacket hitting a barrier… and being reflected:*

*A wavepacket of higher energy hitting a barrier… and being transmitted:*

*A wavepacket incident on a Kronig-Penney crystal:*

### Questions/Feedback

Email us at timothy.atherton@tufts.edu